Problem 49 - Algebra 2

Question

ACT/SAT What is the center of the circle with equation \(x^{2}+y^{2}-10 x+\) \(6 y+27=0 ?\) $$ \begin{array}{l}{\text { A }(-10,6)} \\ {\text { B }(1,1)} \\ {\text { C }(10,-6)} \\ {\text { D }(5,-3)}\end{array} $$

Step-by-Step Solution

Answer

The center is (5, -3), which is option D.

Step 1: Identify the Circle Equation

The given equation of the circle is \(x^2 + y^2 - 10x + 6y + 27 = 0\). To find the center of the circle from this equation, we need to rewrite it in the standard form of a circle equation.

Step 2: Rewrite Equation in Standard Form

The standard form of a circle equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center. To convert the given equation into this form, we need to complete the square for the \(x\) and \(y\) terms.

Step 3: Complete the Square for x

To complete the square for \(x^2 - 10x\), take half of the coefficient of \(x\), square it, and add it inside the equation. \((-10/2)^2 = 25\), so we add and subtract 25: \[ x^2 - 10x + 25 - 25 \]

Step 4: Complete the Square for y

Similarly, complete the square for \(y^2 + 6y\) by taking half of the coefficient of \(y\), squaring it, and adding it inside the equation. \((6/2)^2 = 9\), so we add and subtract 9: \[ y^2 + 6y + 9 - 9 \]

Step 5: Rewrite Entire Equation

Substitute the completed squares back into the equation and reorganize: \[ (x^2 - 10x + 25) + (y^2 + 6y + 9) = -27 + 25 + 9 \] This simplifies to: \[ (x - 5)^2 + (y + 3)^2 = 7 \]

Step 6: Determine Circle's Center

From \((x - 5)^2 + (y + 3)^2 = 7\), the center of the circle \((h, k)\) is \((5, -3)\).

Step 7: Verify the Answer with Options

Check which option matches the center \((5, -3)\): - A \((-10, 6)\)- B \((1, 1)\)- C \((10, -6)\)- D \((5, -3)\) Thus, the correct choice is option D.

Key Concepts

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This method is handy when dealing with equations of a circle, as it helps in restructuring the equation into the standard form. To complete the square for a term like \(x^2 - 10x\):

• First, take half of the coefficient of \(x\). Here, half of \(-10\) is \(-5\).
• Next, square this result. So, \((-5)^2 = 25\).
• Finally, add and subtract this squared value inside the equation, turning \(x^2 - 10x\) into \((x - 5)^2 - 25\).

This process reconfigures the expression into a more manageable form, making it easier to identify the circle's attributes. The same steps apply for the \(y\) terms. Complete these squares to transform the original equation fully, allowing that transition to the circle's standard form.

Center of a Circle

The center of a circle is essentially the point from which all points on the circle are equidistant. For any circle represented by the equation in the standard form \((x - h)^2 + (y - k)^2 = r^2\), the center of the circle is denoted by \((h, k)\).To determine this center from any given equation, like the one from the exercise:

• First, rearrange the equation so it resembles the standard form by completing the square, as described earlier.
• Identify the values of \(h\) and \(k\) from the resulting expressions \((x - h)^2\) and \((y - k)^2\).
• For our problem, after completing the square, we have \((x - 5)^2 + (y + 3)^2\). Thus, \(h = 5\) and \(k = -3\).

These values point to \((5, -3)\) being the center, making it crucial to transform the equation correctly to ensure the center is accurately identified.

Standard Form of a Circle Equation

The standard form of a circle equation is a concise way of expressing the essential characteristics of a circle. This is given by \[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) is the center of the circle, and \(r\) represents the radius.

• To reach this form starting from a general quadratic equation, you must complete the square for both the \(x\) and \(y\) terms.
• After completing the square, reorganize the equation so that the completed squares are on one side, and numerical constants are on the other.
• For instance, in the given exercise, after using complete the square on both variables, the equation becomes \((x - 5)^2 + (y + 3)^2 = 7\).

This transformation simplifies identifying circle parameters and is vital in analytical geometry for understanding circles' behavior in a coordinate plane.